A305201 Expansion of e.g.f. Product_{k>=1} 1/(1 - H(k)*x^k), where H(k) is the k-th harmonic number.

1, 1, 5, 26, 208, 1644, 18728, 201466, 2809672, 39505800, 647509992, 10851033984, 210456343392, 4090234000800, 89123794754304, 2000019423403824, 48674645933985408, 1217362548455301504, 32913123947574009984, 910006995701419453440, 26898048642355515339264

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..430

Vaclav Kotesovec, Graph - the asymptotic ratio (5000 terms)

Formula

E.g.f.: Product_{k>=1} 1/(1 - (A001008(k)/A002805(k))*x^k).

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} H(j)^k*x^(j*k)/k).

a(n) ~ n! * c * (3/2)^(n/2 + 1) / (3 - sqrt(6)), where c = Product_{k>=3} 1/(1 - (2/3)^(k/2) * H(k)) = 20723937.5142714953478411012151498609843924051679047516... - Vaclav Kotesovec, Nov 05 2019

Mathematica

nmax = 20; CoefficientList[Series[Product[1/(1 - HarmonicNumber[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[HarmonicNumber[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d HarmonicNumber[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 20}]

Crossrefs

Cf. A001008, A002805, A007841, A303970, A305203.

Sequence in context: A366215 A348206 A007286 * A099032 A277489 A094652

Adjacent sequences: A305198 A305199 A305200 * A305202 A305203 A305204

Keyword

nonn

Author

Ilya Gutkovskiy, May 27 2018

Status

approved